Problem 5 - College Algebra

Question

In Exercises \(1-8,\) find the domain of each rational function. $$h(x)=\frac{x+7}{x^{2}-49}$$

Step-by-Step Solution

Answer

The domain of the function \(h(x)=\frac{x+7}{x^{2}-49}\) is \((-\infty, -7) \cup (-7, 7) \cup (7, +\infty)\).

Step 1: Identify the Denominator

The denominator of the function \(h(x)\) is \(x^{2}-49\). To avoid a zero denominator, we have to find the values of \(x\) that make the denominator zero. These values will not be included in the domain of \(h(x)\).

Step 2: Solve the Equation

Set the denominator equal to zero and solve for \(x\). This gives us the equation \(x^{2}-49=0\). This can be factored to \((x-7)(x+7)=0\), setting each factor equal to zero gives the solutions \(x=7\) and \(x=-7\). These are the values that make the denominator zero.

Step 3: Find the Domain

The domain of \(h(x)\) then consists of all real numbers except \(x=7\) and \(x=-7\). In interval notation, this domain can be written as \((-\infty, -7) \cup (-7, 7) \cup (7, +\infty)\).

Key Concepts

Zero Denominator

When working with rational functions, it's crucial to understand the concept of a zero denominator. In mathematics, division by zero is undefined because it doesn't have a meaningful value — it's impossible to divide something into zero groups. Therefore, when dealing with functions like
\(h(x) = \frac{x+7}{x^2 - 49}\),
we must ensure the denominator never equals zero. To do this, we find the values of x that would cause the denominator to be zero and exclude them from the domain of the function. These are known as excluded values. Ensuring a non-zero denominator is key to defining the domain accurately, as we don't want to include values that would lead to an undefined expression.

Solving Quadratic Equations

When identifying the zero denominator, we often come across quadratic equations. A quadratic equation is any equation that can be rearranged into the standard form
\(ax^2 + bx + c = 0\),
where a, b, and c are constants, and a \(eq\) 0. To find the values that result in a zero denominator, we set the quadratic equation from the denominator to zero and solve for x. In our case,
\(x^2 - 49 = 0\),
is solved by factoring the difference of squares into
\((x - 7)(x + 7) = 0\).
We then set each factor equal to zero to find the values of x which are \(x = 7\) and \(x = -7\). This method of solving quadratics is a fundamental skill in algebra.

Interval Notation

Once we have determined which values to exclude, we can express the domain using interval notation. This is a way of writing subsets of the real number line. An interval notation consists of a pair of numbers that define the endpoints of an interval, along with parentheses or brackets to denote whether these endpoints are excluded or included, respectively.
For our function
\(h(x) = \frac{x+7}{x^2 - 49}\),
the domain excludes \(x = 7\) and \(x = -7\). Thus, we write the domain as
\((-\infty, -7) \cup (-7, 7) \cup (7, +\infty)\),
indicating that all real numbers are included except \(-7\) and \(7\), which are excluded due to the parentheses. Using interval notation helps us state the domain clearly and concisely.

Excluded Values

The excluded values in a function's domain are the values of the variable for which the function is not defined. In the case of rational functions, these values are usually the result of zero denominators.
For the given function
\(h(x) = \frac{x+7}{x^2 - 49}\),
the excluded values are \(7\) and \(-7\), as those make the denominator equal to zero. Identifying excluded values is an essential step in finding the domain since it ensures all the function's values are defined. It is also important when graphing the function, as these points will usually correspond to vertical asymptotes, which are lines that the graph of the function will approach but never touch.